while the squared chordal distance between the two subspaces \([{\bf X}_i]\)  and \([{\bf X}_j]\)  is given by \(\sum _{i=1}^M \sin ^2 \theta _i\) .
The sum in (REF ) is a pairwise interaction energy in the spirit of the well–studied Riesz or logarithmic energies of importance in Potential Theory (see [1]} for a complete monograph dedicated to energy minimization in the sphere and other spaces). We refer to the function (again, choosing representatives \({\bf A}\)  and \({\bf B}\)  such that \({\bf A}^H{\bf A}={\bf B}^H{\bf B}={\bf I}_M\) )
\([{\bf A}],[{\bf B}]\in \mathbb {G}r(M,{\mathbb {C}}^{T})\mapsto \operatorname{det}( {\bf I}_M -{\bf A}^H{\bf B}{\bf B}^H {\bf A}) =\operatorname{det}( {\bf I}_M -{\bf B}^H{\bf A}{\bf A}^H {\bf B}) ,\) 
